Research Papers

On the Solutions of Interval Systems for Under-Constrained and Redundant Parallel Manipulators

[+] Author and Article Information
Leila Notash

Department of Mechanical and
Materials Engineering,
Queen's University,
Kingston, ON K7L 3N6, Canada
e-mail: leila.notash@queensu.ca

Manuscript received April 12, 2017; final manuscript received August 23, 2017; published online September 26, 2017. Assoc. Editor: Raffaele Di Gregorio.

J. Mechanisms Robotics 9(6), 061009 (Sep 26, 2017) (9 pages) Paper No: JMR-17-1107; doi: 10.1115/1.4037803 History: Received April 12, 2017; Revised August 23, 2017

For under-constrained and redundant parallel manipulators, the actuator inputs are studied with bounded variations in parameters and data. Problem is formulated within the context of force analysis. Discrete and analytical methods for interval linear systems are presented, categorized, and implemented to identify the solution set, as well as the minimum 2-norm least-squares solution set. The notions of parameter dependency and solution subsets are considered. The hyperplanes that bound the solution in each orthant characterize the solution set of manipulators. While the parameterized form of the interval entries of the Jacobian matrix and wrench produce the minimum 2-norm least-squares solution for the under-constrained and over-constrained systems of real matrices and vectors within the interval Jacobian matrix and wrench vector, respectively. Example manipulators are used to present the application of methods for identifying the solution and minimum norm solution sets for actuator forces/torques.

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Hansen, E. , 1992, Global Optimization Using Interval Analysis, 1st ed., Marcel Dekker, New York.
Moore, R. , Kearfott, R. B. , and Cloud, M. J. , 2009, Introduction to Interval Analysis, SIAM, Philadelphia, PA. [CrossRef]
Kearfott, R. B. , 1996, Rigorous Global Search: Continuous Problems, Kluwer Academic, Dordrecht, The Netherlands. [CrossRef]
Rump, S. , 1983, Solving Algebraic Problems With High Accuracy, A New Approach in Scientific Computation, U. Kulisch , and W. Miranker , eds., Academic Press, San Diego, CA, pp. 51–120. [CrossRef]
Rohn, J. , 1996, “ Enclosing Solutions of Overdetermined Systems of Linear Interval Equations,” Reliab. Comput., 2(2), pp. 167–171. [CrossRef]
Bentbib, A. H. , 2002, “ Solving the Full Rank Interval Least Squares Problem,” Appl. Numer. Math., 41(2), pp. 283–294. [CrossRef]
Hansen, E. , and Walster, G. W. , 2006, “ Solving Overdetermined Systems of Interval Linear Equations,” Reliab. Comput., 12(3), pp. 239–243. [CrossRef]
Hladík, M. , 2012, “ Enclosures for the Solution Set of Parametric Interval Linear Systems,” Int. J. Appl. Math. Comput. Sci., 22(3), pp. 561–574. [CrossRef]
Horáček, J. , and Hladík, M. , 2013, “ Subsquares Approach—A Simple Scheme for Solving Overdetermined Interval Linear Systems,” Parallel Processing and Applied Mathematics, R. Wyrzykowski , et al., eds., Springer, Berlin, pp. 613–622. [CrossRef]
Notash, L. , 2015, “ Analytical Methods for Solution Sets of Interval Wrench,” ASME Paper No. DETC2015-47575.
Hartfiel, D. , 1980, “ Concerning the Solution Set of Ax = b Where PAQ and pbq,” Numer. Math., 35(3), pp. 355–359. [CrossRef]
Popova, E. D. , and Krämer, W. , 2008, “ Visualizing Parametric Solution Sets,” BIT Numer. Math., 48(1), pp. 95–115. [CrossRef]
Nazari, V. , and Notash, L. , 2016, “ Motion Analysis of Manipulators With Uncertainty in Kinematic Parameters,” ASME J. Mech. Rob., 8(2), p. 021014. [CrossRef]
Notash, L. , 2012, “ A Methodology for Actuator Failure Recovery in Parallel Manipulators,” Mech. Mach. Theory, 46(4), pp. 454–465. [CrossRef]
Fieldler, M. , Nedoma, J. , Ramik, J. , and Rohn, J. , 2006, Linear Optimization Problems With Inexact Data, Springer, New York.
Notash, L. , 2016, “ On the Solution Set for Positive Wire Tension With Uncertainty in Wire-Actuated Parallel Manipulators,” ASME J. Mech. Rob., 8(4), p. 044506. [CrossRef]
Notash, L. , 2017, “ Wrench Accuracy for Parallel Manipulators and Interval Dependency,” ASME J. Mech. Rob., 9(1), p. 011008. [CrossRef]
Oettli, W. , 1965, “ On the Solution Set of a Linear System With Inaccurate Coefficients,” SIAM J. Numer. Anal., Ser. B, 2(1), pp. 115–118.
Notash, L. , 2017, “ Solutions of Interval Systems for Under-Constrained and Redundant Parallel Manipulators,” ASME Paper No. DETC2017-67091.
Rump, S. M. , 1999, INTLAB—INTerval LABoratory, Developments in Reliable Computing, T. Csendes , ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 77–104. [CrossRef]


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Fig. 1

Parallel manipulators with (a) two actuators and (b) three actuators

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Fig. 2

Enclosure and subsets of solution set

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Fig. 3

Parameters of planar manipulators

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Fig. 4

2-norm of midpoint and radius of residual vector: (a) in solution (b) outside solution set

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Fig. 5

Solution and minimum norm solution using (a) rays, (b) six, (c) two and (d) seven parameters

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Fig. 6

Solution subsets (a) real F and (b) interval F

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Fig. 7

Solution set using closed half-planes

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Fig. 8

Solution and minimum norm solution sets using (a) six parameters and (b) two parameters

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Fig. 9

Minimum norm solution set using eight independent parameters

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Fig. 10

Solution and minimum norm solution sets for interval F

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Fig. 11

Solution subsets (a) real F and (b) interval F




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